3.198 \(\int \frac{1}{\left (d+e x^2\right )^2 \left (d^2-e^2 x^4\right )} \, dx\)

Optimal. Leaf size=89 \[ \frac{7 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{16 d^{7/2} \sqrt{e}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{7/2} \sqrt{e}}+\frac{5 x}{16 d^3 \left (d+e x^2\right )}+\frac{x}{8 d^2 \left (d+e x^2\right )^2} \]

[Out]

x/(8*d^2*(d + e*x^2)^2) + (5*x)/(16*d^3*(d + e*x^2)) + (7*ArcTan[(Sqrt[e]*x)/Sqr
t[d]])/(16*d^(7/2)*Sqrt[e]) + ArcTanh[(Sqrt[e]*x)/Sqrt[d]]/(8*d^(7/2)*Sqrt[e])

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Rubi [A]  time = 0.211968, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{7 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{16 d^{7/2} \sqrt{e}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{7/2} \sqrt{e}}+\frac{5 x}{16 d^3 \left (d+e x^2\right )}+\frac{x}{8 d^2 \left (d+e x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x^2)^2*(d^2 - e^2*x^4)),x]

[Out]

x/(8*d^2*(d + e*x^2)^2) + (5*x)/(16*d^3*(d + e*x^2)) + (7*ArcTan[(Sqrt[e]*x)/Sqr
t[d]])/(16*d^(7/2)*Sqrt[e]) + ArcTanh[(Sqrt[e]*x)/Sqrt[d]]/(8*d^(7/2)*Sqrt[e])

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Rubi in Sympy [A]  time = 58.1471, size = 82, normalized size = 0.92 \[ \frac{x}{8 d^{2} \left (d + e x^{2}\right )^{2}} + \frac{5 x}{16 d^{3} \left (d + e x^{2}\right )} + \frac{7 \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{16 d^{\frac{7}{2}} \sqrt{e}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{8 d^{\frac{7}{2}} \sqrt{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x**2+d)**2/(-e**2*x**4+d**2),x)

[Out]

x/(8*d**2*(d + e*x**2)**2) + 5*x/(16*d**3*(d + e*x**2)) + 7*atan(sqrt(e)*x/sqrt(
d))/(16*d**(7/2)*sqrt(e)) + atanh(sqrt(e)*x/sqrt(d))/(8*d**(7/2)*sqrt(e))

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Mathematica [A]  time = 0.100333, size = 76, normalized size = 0.85 \[ \frac{\frac{\sqrt{d} x \left (7 d+5 e x^2\right )}{\left (d+e x^2\right )^2}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}}{16 d^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x^2)^2*(d^2 - e^2*x^4)),x]

[Out]

((Sqrt[d]*x*(7*d + 5*e*x^2))/(d + e*x^2)^2 + (7*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqr
t[e] + (2*ArcTanh[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e])/(16*d^(7/2))

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Maple [A]  time = 0.015, size = 73, normalized size = 0.8 \[{\frac{1}{8\,{d}^{3}}{\it Artanh} \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{5\,e{x}^{3}}{16\,{d}^{3} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{7\,x}{16\,{d}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{7}{16\,{d}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x^2+d)^2/(-e^2*x^4+d^2),x)

[Out]

1/8/d^3/(d*e)^(1/2)*arctanh(x*e/(d*e)^(1/2))+5/16/d^3/(e*x^2+d)^2*x^3*e+7/16*x/d
^2/(e*x^2+d)^2+7/16/d^3/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((e^2*x^4 - d^2)*(e*x^2 + d)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.295003, size = 1, normalized size = 0.01 \[ \left [\frac{7 \,{\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) +{\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \log \left (\frac{2 \, d e x +{\left (e x^{2} + d\right )} \sqrt{d e}}{e x^{2} - d}\right ) +{\left (5 \, e x^{3} + 7 \, d x\right )} \sqrt{d e}}{16 \,{\left (d^{3} e^{2} x^{4} + 2 \, d^{4} e x^{2} + d^{5}\right )} \sqrt{d e}}, \frac{4 \,{\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \arctan \left (\frac{\sqrt{-d e} x}{d}\right ) + 7 \,{\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \log \left (\frac{2 \, d e x +{\left (e x^{2} - d\right )} \sqrt{-d e}}{e x^{2} + d}\right ) + 2 \,{\left (5 \, e x^{3} + 7 \, d x\right )} \sqrt{-d e}}{32 \,{\left (d^{3} e^{2} x^{4} + 2 \, d^{4} e x^{2} + d^{5}\right )} \sqrt{-d e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((e^2*x^4 - d^2)*(e*x^2 + d)^2),x, algorithm="fricas")

[Out]

[1/16*(7*(e^2*x^4 + 2*d*e*x^2 + d^2)*arctan(sqrt(d*e)*x/d) + (e^2*x^4 + 2*d*e*x^
2 + d^2)*log((2*d*e*x + (e*x^2 + d)*sqrt(d*e))/(e*x^2 - d)) + (5*e*x^3 + 7*d*x)*
sqrt(d*e))/((d^3*e^2*x^4 + 2*d^4*e*x^2 + d^5)*sqrt(d*e)), 1/32*(4*(e^2*x^4 + 2*d
*e*x^2 + d^2)*arctan(sqrt(-d*e)*x/d) + 7*(e^2*x^4 + 2*d*e*x^2 + d^2)*log((2*d*e*
x + (e*x^2 - d)*sqrt(-d*e))/(e*x^2 + d)) + 2*(5*e*x^3 + 7*d*x)*sqrt(-d*e))/((d^3
*e^2*x^4 + 2*d^4*e*x^2 + d^5)*sqrt(-d*e))]

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Sympy [A]  time = 3.23939, size = 255, normalized size = 2.87 \[ - \frac{\sqrt{\frac{1}{d^{7} e}} \log{\left (- \frac{20 d^{11} e \left (\frac{1}{d^{7} e}\right )^{\frac{3}{2}}}{371} - \frac{351 d^{4} \sqrt{\frac{1}{d^{7} e}}}{371} + x \right )}}{16} + \frac{\sqrt{\frac{1}{d^{7} e}} \log{\left (\frac{20 d^{11} e \left (\frac{1}{d^{7} e}\right )^{\frac{3}{2}}}{371} + \frac{351 d^{4} \sqrt{\frac{1}{d^{7} e}}}{371} + x \right )}}{16} - \frac{7 \sqrt{- \frac{1}{d^{7} e}} \log{\left (- \frac{245 d^{11} e \left (- \frac{1}{d^{7} e}\right )^{\frac{3}{2}}}{106} - \frac{351 d^{4} \sqrt{- \frac{1}{d^{7} e}}}{106} + x \right )}}{32} + \frac{7 \sqrt{- \frac{1}{d^{7} e}} \log{\left (\frac{245 d^{11} e \left (- \frac{1}{d^{7} e}\right )^{\frac{3}{2}}}{106} + \frac{351 d^{4} \sqrt{- \frac{1}{d^{7} e}}}{106} + x \right )}}{32} + \frac{7 d x + 5 e x^{3}}{16 d^{5} + 32 d^{4} e x^{2} + 16 d^{3} e^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x**2+d)**2/(-e**2*x**4+d**2),x)

[Out]

-sqrt(1/(d**7*e))*log(-20*d**11*e*(1/(d**7*e))**(3/2)/371 - 351*d**4*sqrt(1/(d**
7*e))/371 + x)/16 + sqrt(1/(d**7*e))*log(20*d**11*e*(1/(d**7*e))**(3/2)/371 + 35
1*d**4*sqrt(1/(d**7*e))/371 + x)/16 - 7*sqrt(-1/(d**7*e))*log(-245*d**11*e*(-1/(
d**7*e))**(3/2)/106 - 351*d**4*sqrt(-1/(d**7*e))/106 + x)/32 + 7*sqrt(-1/(d**7*e
))*log(245*d**11*e*(-1/(d**7*e))**(3/2)/106 + 351*d**4*sqrt(-1/(d**7*e))/106 + x
)/32 + (7*d*x + 5*e*x**3)/(16*d**5 + 32*d**4*e*x**2 + 16*d**3*e**2*x**4)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((e^2*x^4 - d^2)*(e*x^2 + d)^2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError